This note derives the Stark effect on the hydrogen ground state. Since spin is irrelevant for the Stark effect, it will be ignored.
The unperturbed ground state of hydrogen was derived in chapter
4.3. Following the convention in perturbation theory to
append a subscript zero to the unperturbed state, it can be summarized
as:
The Stark perturbation produces a change in this wave
function that satisfies, from (A.243),
Now in polar coordinates, and for the
-derivative of to produce something that is proportional
to , must be proportional to . (The Laplacian
in the second term always produces lower powers of than the
-derivative and can for now be ignored.) So, to balance the
right hand side, should contain a highest power of equal to:
small perturbationbecomes larger than the unperturbed wave function far from the atom because of the growing value of . It is implicitly assumed that the electric field terminates before a real problem arises. This is related to the possibility of the electron tunneling out of the atom if the potential far from the atom is less than its energy in the atom: if the electron can tunnel out, there is strictly speaking no bound state.)
Now according to (A.243), the second order energy change
can be found as